= Baer sum
{c}
To add two extensions of $M$ by $N$, take their <direct sum>, form a <pullback in a category> along the diagonal $M\to M\oplus M$, and form a <pushout in a category> along addition $N\oplus N\to N$. This produces another extension of $M$ by $N$ and defines addition in $\operatorname{Ext}^1_R(M,N)$. In a <quiver representation>, vertexwise splittings express the two extensions by off-diagonal arrow matrices, and the Baer sum adds those matrices modulo the <extension complex of quiver representations> coboundaries.
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